InductanceConsider 2 coils of wire C2 C1 I1 I2

The magnetic field through C1 is

and similarly, the flux through C2 is

This is important because

When currents change, that will affect those and other currents by “inductance”.

Li = self inductance

(how Ii affects itself)

Mij = mutual inductance (how I

j affects I

i)

Theorem M12 = M21

Proof

B1 I

1

B2 I

2

“Neumann’s equation”

Q. E. D.

Example. The mutual inductance between coaxial solenoids, Sol1 and Sol2, assuming these are densely wound solenoids and Sol2 is outside Sol1 ...

Assume l2 < l1 and A2 > A1.Neglect end effects. Note that M12 = M21.

(2)

Self Inductance (L)

Example: a solenoid (= a wire densely wound around a long cylinder)

Units: Tm/A x m2/m = Tm2/A = H (henry)

LR Circuit

I = emf /R = - d/dt (LI) /RdI /dt = - (R/L) II(t) = I

0 exp(-Rt/L)

Energy density of a magnetic field

Apply energy conservation to the LR circuit. The power dissipated in resistance is P = I2 R (Joule’s law).

Conservation of energy implies

where U is the energy of the magnetic field in the inductor. So

P dt = dQ VP = IV = I2R

x